ToRadicals

ToRadicals[expr]

attempts to express all Root objects in expr in terms of radicals.

Details and Options

  • ToRadicals can always give expressions in terms of radicals when the highest degree of the polynomial that appears in any Root object is four.
  • There are some cases in which expressions involving radicals can in principle be given, but ToRadicals cannot find them.
  • If Root objects in expr contain parameters, ToRadicals[expr] may yield a result that is not equal to expr for all values of the parameters.
  • ToRadicals automatically threads over lists, as well as equations, inequalities, and logic functions.

Examples

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Basic Examples  (1)

Scope  (3)

All cubic Root objects can be converted into radicals:

All quartic Root objects can be converted into radicals:

Some higher‐degree Root objects can be represented in terms of radicals:

ToRadicals also works with AlgebraicNumber objects:

Generalizations & Extensions  (1)

ToRadicals converts trigonometric functions of rational multiples of :

Options  (4)

Cubics  (2)

With Cubics->False the general formulas for solving cubic equations are not used:

Converting some cubic Root objects does not require the general formulas:

Quartics  (2)

With Quartics->False the general formulas for solving quartic equations are not used:

Converting some quartic Root objects does not require the general formulas:

Properties & Relations  (2)

RootReduce[ToRadicals[r]]==r for any algebraic number r given as a Root object:

By default Reduce will not produce radical solutions for general cubics:

Use ToRadicals to convert:

Alternatively set Cubics->True:

Possible Issues  (3)

In this case ToRadicals succeeds on the unreduced expression:

In this case ToRadicals succeeds on the reduced expression:

ToRadicals converts Root objects containing parameters:

The result may not be equal to the Root object for some values of the parameter:

Wolfram Research (1996), ToRadicals, Wolfram Language function, https://reference.wolfram.com/language/ref/ToRadicals.html (updated 2007).

Text

Wolfram Research (1996), ToRadicals, Wolfram Language function, https://reference.wolfram.com/language/ref/ToRadicals.html (updated 2007).

CMS

Wolfram Language. 1996. "ToRadicals." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/ToRadicals.html.

APA

Wolfram Language. (1996). ToRadicals. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ToRadicals.html

BibTeX

@misc{reference.wolfram_2024_toradicals, author="Wolfram Research", title="{ToRadicals}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ToRadicals.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_toradicals, organization={Wolfram Research}, title={ToRadicals}, year={2007}, url={https://reference.wolfram.com/language/ref/ToRadicals.html}, note=[Accessed: 22-December-2024 ]}